sdl-ios-xcode: src/video/e

annotate src/video/e_sqrt.h @ 2661:d38309be5178 gsoc2008_audio_resampling

The windowed sinc filter generation code seems to be working fine. The FIR filtering code is also now working reasonably well. Occasionally the FIR filter will pop, but setting the normalization factor lower seems to help this. I suspect the problem is in the fixed point multiply/add. I also have a hunch the zero stuffing/sample discarding code is not correct, and I'll look at that soon to get it sorted out.

author	Aaron Wishnick <schnarf@gmail.com>
date	Wed, 02 Jul 2008 08:04:50 +0000
parents	c121d94672cb
children

rev	line source
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	1 /* @(#)e_sqrt.c 5.1 93/09/24 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	2 /*
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	3 * ====================================================
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	5 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	6 * Developed at SunPro, a Sun Microsystems, Inc. business.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	7 * Permission to use, copy, modify, and distribute this
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	8 * software is freely granted, provided that this notice
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	9 * is preserved.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	10 * ====================================================
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	11 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	12
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	13 #if defined(LIBM_SCCS) && !defined(lint)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	14 static char rcsid[] = "$NetBSD: e_sqrt.c,v 1.8 1995/05/10 20:46:17 jtc Exp $";
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	15 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	16
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	17 /* __ieee754_sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	18 * Return correctly rounded sqrt.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	19 * ------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	20 * \| Use the hardware sqrt if you have one \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	21 * ------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	22 * Method:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	23 * Bit by bit method using integer arithmetic. (Slow, but portable)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	24 * 1. Normalization
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	25 * Scale x to y in [1,4) with even powers of 2:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	26 * find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	27 * sqrt(x) = 2^k * sqrt(y)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	28 * 2. Bit by bit computation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	29 * Let q = sqrt(y) truncated to i bit after binary point (q = 1),
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	30 * i 0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	31 * i+1 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	32 * s = 2q , and y = 2 ( y - q ). (1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	33 * i i i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	34 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	35 * To compute q from q , one checks whether
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	36 * i+1 i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	37 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	38 * -(i+1) 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	39 * (q + 2 ) <= y. (2)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	40 * i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	41 * -(i+1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	42 * If (2) is false, then q = q ; otherwise q = q + 2 .
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	43 * i+1 i i+1 i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	44 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	45 * With some algebric manipulation, it is not difficult to see
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	46 * that (2) is equivalent to
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	47 * -(i+1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	48 * s + 2 <= y (3)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	49 * i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	50 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	51 * The advantage of (3) is that s and y can be computed by
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	52 * i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	53 * the following recurrence formula:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	54 * if (3) is false
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	55 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	56 * s = s , y = y ; (4)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	57 * i+1 i i+1 i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	58 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	59 * otherwise,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	60 * -i -(i+1)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	61 * s = s + 2 , y = y - s - 2 (5)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	62 * i+1 i i+1 i i
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	63 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	64 * One may easily use induction to prove (4) and (5).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	65 * Note. Since the left hand side of (3) contain only i+2 bits,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	66 * it does not necessary to do a full (53-bit) comparison
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	67 * in (3).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	68 * 3. Final rounding
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	69 * After generating the 53 bits result, we compute one more bit.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	70 * Together with the remainder, we can decide whether the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	71 * result is exact, bigger than 1/2ulp, or less than 1/2ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	72 * (it will never equal to 1/2ulp).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	73 * The rounding mode can be detected by checking whether
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	74 * huge + tiny is equal to huge, and whether huge - tiny is
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	75 * equal to huge for some floating point number "huge" and "tiny".
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	76 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	77 * Special cases:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	78 * sqrt(+-0) = +-0 ... exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	79 * sqrt(inf) = inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	80 * sqrt(-ve) = NaN ... with invalid signal
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	81 * sqrt(NaN) = NaN ... with invalid signal for signaling NaN
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	82 *
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	83 * Other methods : see the appended file at the end of the program below.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	84 *---------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	85 */
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	86
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	87 /#include "math.h"/
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	88 #include "math_private.h"
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	89
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	90 #ifdef __STDC__
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	91 double SDL_NAME(copysign) (double x, double y)
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	92 #else
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	93 double SDL_NAME(copysign) (x, y)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	94 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	95 x,
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	96 y;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	97 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	98 {
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	99 u_int32_t hx, hy;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	100 GET_HIGH_WORD(hx, x);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	101 GET_HIGH_WORD(hy, y);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	102 SET_HIGH_WORD(x, (hx & 0x7fffffff) \| (hy & 0x80000000));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	103 return x;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	104 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	105
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	106 #ifdef __STDC__
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	107 double SDL_NAME(scalbn) (double x, int n)
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	108 #else
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	109 double SDL_NAME(scalbn) (x, n)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	110 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	111 x;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	112 int
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	113 n;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	114 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	115 {
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	116 int32_t k, hx, lx;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	117 EXTRACT_WORDS(hx, lx, x);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	118 k = (hx & 0x7ff00000) >> 20; /* extract exponent */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	119 if (k == 0) { /* 0 or subnormal x */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	120 if ((lx \| (hx & 0x7fffffff)) == 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	121 return x; /* +-0 */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	122 x *= two54;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	123 GET_HIGH_WORD(hx, x);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	124 k = ((hx & 0x7ff00000) >> 20) - 54;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	125 if (n < -50000)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	126 return tiny * x; /underflow /
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	127 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	128 if (k == 0x7ff)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	129 return x + x; /* NaN or Inf */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	130 k = k + n;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	131 if (k > 0x7fe)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	132 return huge * SDL_NAME(copysign) (huge, x); /* overflow */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	133 if (k > 0) { /* normal result */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	134 SET_HIGH_WORD(x, (hx & 0x800fffff) \| (k << 20));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	135 return x;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	136 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	137 if (k <= -54) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	138 if (n > 50000) /* in case integer overflow in n+k */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	139 return huge * SDL_NAME(copysign) (huge, x); /overflow /
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	140 else
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	141 return tiny * SDL_NAME(copysign) (tiny, x); /underflow /
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	142 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	143 k += 54; /* subnormal result */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	144 SET_HIGH_WORD(x, (hx & 0x800fffff) \| (k << 20));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	145 return x * twom54;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	146 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	147
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	148 #ifdef __STDC__
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	149 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	150 __ieee754_sqrt(double x)
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	151 #else
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	152 double
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	153 __ieee754_sqrt(x)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	154 double x;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	155 #endif
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	156 {
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	157 double z;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	158 int32_t sign = (int) 0x80000000;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	159 int32_t ix0, s0, q, m, t, i;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	160 u_int32_t r, t1, s1, ix1, q1;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	161
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	162 EXTRACT_WORDS(ix0, ix1, x);
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	163
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	164 /* take care of Inf and NaN */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	165 if ((ix0 & 0x7ff00000) == 0x7ff00000) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	166 return x * x + x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	167 sqrt(-inf)=sNaN */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	168 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	169 /* take care of zero */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	170 if (ix0 <= 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	171 if (((ix0 & (~sign)) \| ix1) == 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	172 return x; /* sqrt(+-0) = +-0 */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	173 else if (ix0 < 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	174 return (x - x) / (x - x); /* sqrt(-ve) = sNaN */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	175 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	176 /* normalize x */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	177 m = (ix0 >> 20);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	178 if (m == 0) { /* subnormal x */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	179 while (ix0 == 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	180 m -= 21;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	181 ix0 \|= (ix1 >> 11);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	182 ix1 <<= 21;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	183 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	184 for (i = 0; (ix0 & 0x00100000) == 0; i++)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	185 ix0 <<= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	186 m -= i - 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	187 ix0 \|= (ix1 >> (32 - i));
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	188 ix1 <<= i;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	189 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	190 m -= 1023; /* unbias exponent */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	191 ix0 = (ix0 & 0x000fffff) \| 0x00100000;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	192 if (m & 1) { /* odd m, double x to make it even */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	193 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	194 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	195 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	196 m >>= 1; /* m = [m/2] */
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	197
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	198 /* generate sqrt(x) bit by bit */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	199 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	200 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	201 q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	202 r = 0x00200000; /* r = moving bit from right to left */
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	203
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	204 while (r != 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	205 t = s0 + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	206 if (t <= ix0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	207 s0 = t + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	208 ix0 -= t;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	209 q += r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	210 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	211 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	212 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	213 r >>= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	214 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	215
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	216 r = sign;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	217 while (r != 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	218 t1 = s1 + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	219 t = s0;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	220 if ((t < ix0) \|\| ((t == ix0) && (t1 <= ix1))) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	221 s1 = t1 + r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	222 if (((int32_t) (t1 & sign) == sign) && (s1 & sign) == 0)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	223 s0 += 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	224 ix0 -= t;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	225 if (ix1 < t1)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	226 ix0 -= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	227 ix1 -= t1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	228 q1 += r;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	229 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	230 ix0 += ix0 + ((ix1 & sign) >> 31);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	231 ix1 += ix1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	232 r >>= 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	233 }
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	234
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	235 /* use floating add to find out rounding direction */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	236 if ((ix0 \| ix1) != 0) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	237 z = one - tiny; /* trigger inexact flag */
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	238 if (z >= one) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	239 z = one + tiny;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	240 if (q1 == (u_int32_t) 0xffffffff) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	241 q1 = 0;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	242 q += 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	243 } else if (z > one) {
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	244 if (q1 == (u_int32_t) 0xfffffffe)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	245 q += 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	246 q1 += 2;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	247 } else
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	248 q1 += (q1 & 1);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	249 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	250 }
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	251 ix0 = (q >> 1) + 0x3fe00000;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	252 ix1 = q1 >> 1;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	253 if ((q & 1) == 1)
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	254 ix1 \|= sign;
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	255 ix0 += (m << 20);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	256 INSERT_WORDS(z, ix0, ix1);
c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	257 return z;
1330 450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	258 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	259
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	260 /*
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	261 Other methods (use floating-point arithmetic)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	262 -------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	263 (This is a copy of a drafted paper by Prof W. Kahan
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	264 and K.C. Ng, written in May, 1986)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	265
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	266 Two algorithms are given here to implement sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	267 (IEEE double precision arithmetic) in software.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	268 Both supply sqrt(x) correctly rounded. The first algorithm (in
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	269 Section A) uses newton iterations and involves four divisions.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	270 The second one uses reciproot iterations to avoid division, but
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	271 requires more multiplications. Both algorithms need the ability
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	272 to chop results of arithmetic operations instead of round them,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	273 and the INEXACT flag to indicate when an arithmetic operation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	274 is executed exactly with no roundoff error, all part of the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	275 standard (IEEE 754-1985). The ability to perform shift, add,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	276 subtract and logical AND operations upon 32-bit words is needed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	277 too, though not part of the standard.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	278
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	279 A. sqrt(x) by Newton Iteration
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	280
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	281 (1) Initial approximation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	282
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	283 Let x0 and x1 be the leading and the trailing 32-bit words of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	284 a floating point number x (in IEEE double format) respectively
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	285
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	286 1 11 52 ...widths
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	287 ------------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	288 x: \|s\| e \| f \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	289 ------------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	290 msb lsb msb lsb ...order
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	291
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	292
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	293 ------------------------ ------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	294 x0: \|s\| e \| f1 \| x1: \| f2 \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	295 ------------------------ ------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	296
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	297 By performing shifts and subtracts on x0 and x1 (both regarded
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	298 as integers), we obtain an 8-bit approximation of sqrt(x) as
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	299 follows.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	300
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	301 k := (x0>>1) + 0x1ff80000;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	302 y0 := k - T1[31&(k>>15)]. ... y ~ sqrt(x) to 8 bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	303 Here k is a 32-bit integer and T1[] is an integer array containing
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	304 correction terms. Now magically the floating value of y (y's
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	305 leading 32-bit word is y0, the value of its trailing word is 0)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	306 approximates sqrt(x) to almost 8-bit.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	307
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	308 Value of T1:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	309 static int T1[32]= {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	310 0, 1024, 3062, 5746, 9193, 13348, 18162, 23592,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	311 29598, 36145, 43202, 50740, 58733, 67158, 75992, 85215,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	312 83599, 71378, 60428, 50647, 41945, 34246, 27478, 21581,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	313 16499, 12183, 8588, 5674, 3403, 1742, 661, 130,};
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	314
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	315 (2) Iterative refinement
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	316
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	317 Apply Heron's rule three times to y, we have y approximates
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	318 sqrt(x) to within 1 ulp (Unit in the Last Place):
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	319
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	320 y := (y+x/y)/2 ... almost 17 sig. bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	321 y := (y+x/y)/2 ... almost 35 sig. bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	322 y := y-(y-x/y)/2 ... within 1 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	323
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	324
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	325 Remark 1.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	326 Another way to improve y to within 1 ulp is:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	327
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	328 y := (y+x/y) ... almost 17 sig. bits to 2*sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	329 y := y - 0x00100006 ... almost 18 sig. bits to sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	330
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	331 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	332 (x-y )*y
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	333 y := y + 2* ---------- ...within 1 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	334 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	335 3y + x
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	336
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	337
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	338 This formula has one division fewer than the one above; however,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	339 it requires more multiplications and additions. Also x must be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	340 scaled in advance to avoid spurious overflow in evaluating the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	341 expression 3y*y+x. Hence it is not recommended uless division
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	342 is slow. If division is very slow, then one should use the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	343 reciproot algorithm given in section B.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	344
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	345 (3) Final adjustment
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	346
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	347 By twiddling y's last bit it is possible to force y to be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	348 correctly rounded according to the prevailing rounding mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	349 as follows. Let r and i be copies of the rounding mode and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	350 inexact flag before entering the square root program. Also we
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	351 use the expression y+-ulp for the next representable floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	352 numbers (up and down) of y. Note that y+-ulp = either fixed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	353 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	354 mode.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	355
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	356 I := FALSE; ... reset INEXACT flag I
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	357 R := RZ; ... set rounding mode to round-toward-zero
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	358 z := x/y; ... chopped quotient, possibly inexact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	359 If(not I) then { ... if the quotient is exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	360 if(z=y) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	361 I := i; ... restore inexact flag
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	362 R := r; ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	363 return sqrt(x):=y.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	364 } else {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	365 z := z - ulp; ... special rounding
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	366 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	367 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	368 i := TRUE; ... sqrt(x) is inexact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	369 If (r=RN) then z=z+ulp ... rounded-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	370 If (r=RP) then { ... round-toward-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	371 y = y+ulp; z=z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	372 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	373 y := y+z; ... chopped sum
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	374 y0:=y0-0x00100000; ... y := y/2 is correctly rounded.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	375 I := i; ... restore inexact flag
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	376 R := r; ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	377 return sqrt(x):=y.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	378
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	379 (4) Special cases
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	380
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	381 Square root of +inf, +-0, or NaN is itself;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	382 Square root of a negative number is NaN with invalid signal.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	383
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	384
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	385 B. sqrt(x) by Reciproot Iteration
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	386
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	387 (1) Initial approximation
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	388
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	389 Let x0 and x1 be the leading and the trailing 32-bit words of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	390 a floating point number x (in IEEE double format) respectively
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	391 (see section A). By performing shifs and subtracts on x0 and y0,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	392 we obtain a 7.8-bit approximation of 1/sqrt(x) as follows.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	393
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	394 k := 0x5fe80000 - (x0>>1);
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	395 y0:= k - T2[63&(k>>14)]. ... y ~ 1/sqrt(x) to 7.8 bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	396
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	397 Here k is a 32-bit integer and T2[] is an integer array
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	398 containing correction terms. Now magically the floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	399 value of y (y's leading 32-bit word is y0, the value of
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	400 its trailing word y1 is set to zero) approximates 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	401 to almost 7.8-bit.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	402
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	403 Value of T2:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	404 static int T2[64]= {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	405 0x1500, 0x2ef8, 0x4d67, 0x6b02, 0x87be, 0xa395, 0xbe7a, 0xd866,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	406 0xf14a, 0x1091b,0x11fcd,0x13552,0x14999,0x15c98,0x16e34,0x17e5f,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	407 0x18d03,0x19a01,0x1a545,0x1ae8a,0x1b5c4,0x1bb01,0x1bfde,0x1c28d,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	408 0x1c2de,0x1c0db,0x1ba73,0x1b11c,0x1a4b5,0x1953d,0x18266,0x16be0,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	409 0x1683e,0x179d8,0x18a4d,0x19992,0x1a789,0x1b445,0x1bf61,0x1c989,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	410 0x1d16d,0x1d77b,0x1dddf,0x1e2ad,0x1e5bf,0x1e6e8,0x1e654,0x1e3cd,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	411 0x1df2a,0x1d635,0x1cb16,0x1be2c,0x1ae4e,0x19bde,0x1868e,0x16e2e,
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	412 0x1527f,0x1334a,0x11051,0xe951, 0xbe01, 0x8e0d, 0x5924, 0x1edd,};
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	413
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	414 (2) Iterative refinement
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	415
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	416 Apply Reciproot iteration three times to y and multiply the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	417 result by x to get an approximation z that matches sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	418 to about 1 ulp. To be exact, we will have
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	419 -1ulp < sqrt(x)-z<1.0625ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	420
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	421 ... set rounding mode to Round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	422 y := y(1.5-0.5xyy) ... almost 15 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	423 y := y((1.5-2^-30)+0.5xyy)... about 29 sig. bits to 1/sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	424 ... special arrangement for better accuracy
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	425 z := xy ... 29 bits to sqrt(x), with zy<1
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	426 z := z + 0.5z(1-z*y) ... about 1 ulp to sqrt(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	427
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	428 Remark 2. The constant 1.5-2^-30 is chosen to bias the error so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	429 (a) the term z*y in the final iteration is always less than 1;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	430 (b) the error in the final result is biased upward so that
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	431 -1 ulp < sqrt(x) - z < 1.0625 ulp
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	432 instead of \|sqrt(x)-z\|<1.03125ulp.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	433
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	434 (3) Final adjustment
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	435
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	436 By twiddling y's last bit it is possible to force y to be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	437 correctly rounded according to the prevailing rounding mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	438 as follows. Let r and i be copies of the rounding mode and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	439 inexact flag before entering the square root program. Also we
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	440 use the expression y+-ulp for the next representable floating
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	441 numbers (up and down) of y. Note that y+-ulp = either fixed
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	442 point y+-1, or multiply y by nextafter(1,+-inf) in chopped
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	443 mode.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	444
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	445 R := RZ; ... set rounding mode to round-toward-zero
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	446 switch(r) {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	447 case RN: ... round-to-nearest
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	448 if(x<= z*(z-ulp)...chopped) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	449 if(x<= z*(z+ulp)...chopped) z = z; else z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	450 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	451 case RZ:case RM: ... round-to-zero or round-to--inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	452 R:=RP; ... reset rounding mod to round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	453 if(x<z*z ... rounded up) z = z - ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	454 if(x>=(z+ulp)*(z+ulp) ...rounded up) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	455 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	456 case RP: ... round-to-+inf
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	457 if(x>(z+ulp)(z+ulp)...chopped) z = z+2ulp; else
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	458 if(x>z*z ...chopped) z = z+ulp;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	459 break;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	460 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	461
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	462 Remark 3. The above comparisons can be done in fixed point. For
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	463 example, to compare x and w=z*z chopped, it suffices to compare
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	464 x1 and w1 (the trailing parts of x and w), regarding them as
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	465 two's complement integers.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	466
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	467 ...Is z an exact square root?
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	468 To determine whether z is an exact square root of x, let z1 be the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	469 trailing part of z, and also let x0 and x1 be the leading and
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	470 trailing parts of x.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	471
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	472 If ((z1&0x03ffffff)!=0) ... not exact if trailing 26 bits of z!=0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	473 I := 1; ... Raise Inexact flag: z is not exact
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	474 else {
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	475 j := 1 - [(x0>>20)&1] ... j = logb(x) mod 2
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	476 k := z1 >> 26; ... get z's 25-th and 26-th
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	477 fraction bits
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	478 I := i or (k&j) or ((k&(j+j+1))!=(x1&3));
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	479 }
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	480 R:= r ... restore rounded mode
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	481 return sqrt(x):=z.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	482
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	483 If multiplication is cheaper then the foregoing red tape, the
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	484 Inexact flag can be evaluated by
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	485
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	486 I := i;
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	487 I := (z*z!=x) or I.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	488
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	489 Note that z*z can overwrite I; this value must be sensed if it is
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	490 True.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	491
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	492 Remark 4. If z*z = x exactly, then bit 25 to bit 0 of z1 must be
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	493 zero.
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	494
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	495 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	496 z1: \| f2 \|
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	497 --------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	498 bit 31 bit 0
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	499
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	500 Further more, bit 27 and 26 of z1, bit 0 and 1 of x1, and the odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	501 or even of logb(x) have the following relations:
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	502
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	503 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	504 bit 27,26 of z1 bit 1,0 of x1 logb(x)
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	505 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	506 00 00 odd and even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	507 01 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	508 10 10 odd
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	509 10 00 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	510 11 01 even
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	511 -------------------------------------------------
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	512
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	513 (4) Special cases (see (4) of Section A).
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	514
450721ad5436 It's now possible to build SDL without any C runtime at all on Windows, Sam Lantinga <slouken@libsdl.org> parents: diff changeset	515 */
1895 c121d94672cb SDL 1.2 is moving to a branch, and SDL 1.3 is becoming the head. Sam Lantinga <slouken@libsdl.org> parents: 1424 diff changeset	516 /* vi: set ts=4 sw=4 expandtab: */

Mercurial > sdl-ios-xcode

annotate src/video/e_sqrt.h @ 2661:d38309be5178 gsoc2008_audio_resampling